SIAM J. MATRIX ANAL. APPL. c 2012 Society for Industrial and Applied Mathematics Vol. 33, No. 3, pp. 1000–1017 NEAREST LINEAR SYSTEMS WITH HIGHLY DEFICIENT REACHABLE SUBSPACES ∗ † EMRE MENGI Abstract. We consider the 2-norm distance τr(A, B) from a linear time-invariant dynamical system (A, B)ofordern to the nearest system (A +ΔA∗,B +ΔB∗) whose reachable subspace is of dimension r<n. We first present a characterization to test whether the reachable sub- space of the system has dimension r, which resembles and can be considered as a generalization of the Popov–Belevitch–Hautus test for controllability. Then, by exploiting this generalized Popov– Belevitch–Hautus characterization, we derive the main result of this paper, which is a singular value optimization characterization for τr(A, B). A numerical technique to solve the derived singular value optimization problems is described. The numerical results on a few examples illustrate the significance of the derived singular value characterization for computational purposes. Key words. optimization of singular values, reachable subspace, distance to uncontrollability, distance to rank deficiency, minimal system, Lipschitz continuous AMS subject classifications. 93B20, 15A18, 90C26 DOI. 10.1137/100787994 1. Introduction. Given a linear time-invariantdynamical system of order n, (1.1)x ˙(t)=Ax(t)+Bu(t) with zero initial conditions, A ∈ Cn×n,andB ∈ C n×m. This work concerns the distance (1.2) τr(A, B) = inf ΔA ΔB :rank(C(A +ΔA, B +ΔB)) ≤ r with C 2 n−1 (A, B)= BABAB ... A B denoting the controllability matrix, whose rank gives the dimension of the reachable subspace of (1.1). In (1.2) and elsewhere · denotes the matrix 2-norm unless otherwise stated. The quantity τr(A, B) can be considered as a gener alization of the distance to uncontrollability [20, 1, 19] τ(A, B) = inf ΔA ΔB] :rank(C(A +ΔA, B +ΔB)) ≤ n − 1 for which the singular value characterization τ(A, B) = inf σn A − λI B λ∈C ∗Received by the editors March 9, 2010; accepted for publication (in revised form) by P. Benner June 25, 2012; published electronically September 20, 2012. This work was supported in part by the European Commision grant PIRG-GA-268355 and the TUB¨ ITAK˙ (the scientific and technological research council of Turkey) carrier grant 109T660. This work was initiated when the author was holding an S.E.W. Assistant Professorship in Mathematics at the University of California at San Diego. Downloaded 04/21/14 to 212.175.32.130. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php http://www.siam.org/journals/simax/33-3/78799.html †Department of Mathematics, Ko¸c University, Rumelifeneri Yolu, 34450 Sariyer-Istanbul, Turkey ([email protected]). 1000 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NEAREST SYSTEMS WITH DEFICIENT REACHABLE SUBSPACES 1001 is derived by Eising [6]. Above and throughout this paper, σj (C) denotes the jth largest singular value of C. Eising’s characterization is a simple consequence of the Popov–Belevitch–Hautus (PBH) test, which establishes the equivalence of the rank- deficiency of C(A, B) and the condition (1.3) ∃λ ∈ C s.t. rank A − λI B ≤ n − 1. Computational techniques based on Eising’s singular value characterization are de- veloped for low precision [2, 13] as well as for high precision [10, 11]. The quantity τr(A, B) provides an upper bound for the distance to nonminimality. The linear system x˙(t)=Ax(t)+Bu(t), (1.4) y(t)=Cx(t)+Du(t) n×n n× m p×n p×m with zero initial conditions, and A ∈ C , B ∈ C , C ∈ C ,andD ∈ C is called nonminimal if there exists a system of smaller order r<nwhich maps the input u(t) to the output y(t) in exactly the same manner as the original system. Let us use the shorthand notation (A, B, C, D) for the system above and pose the problem ΔA ΔB r×r r×m p×r τr(A, B, C) = inf : ∃Ar ∈ C ,Br ∈ C ,Cr ∈ C ΔC 0 (1.5) (A +ΔA, B +ΔB,C +ΔC, D) ≡ (Ar,Br,Cr,D) , ≡ whereweuse(A +ΔA, B +ΔB,C +ΔC, D) (Ar ,Br,Cr,D)todenotethatthe left-hand and right-hand arguments are equivalent systems. It is well known that when the reachable subspace of (1.4) is of dimension r , there is an equivalent system of order r (see Proposition 2.27 and its proof in [5]) implying τr(A, B, C) ≤ τr(A, B). The distance under consideration τr(A, B) is also related to the sensitivity of the Krylov subspaces. In particular, for the special case when B is a column vector, the range of C(A, B) reduces to a Krylov subspace. The distance τr(A, B)canbe viewed as the backward error so that the perturbed Krylov subspace has dimension r. Consequently, it is inherently connected to the condition numbers for Krylov subspaces, which are formalized and studied by Godunov and his co-authors; see, for instance, [3]. This work, however, concerns the more general case when B is not necessarily a column vector, and the quantity underconsideration can be viewed as a backward error for the simultaneous subspace iteration. The outline of this paper is as follows. In section 2 we derive a generalization of the PBH test (1.3) for the condition rank(C(A, B )) ≤ r. (See Theorem 2.1 for the generalized PBH test and Theorem 2.3 for a linearized version.) In section 3 by exploiting this generalized PBH test we obtain the main result of this paper, which is a singular value optimization characterization for the distance τr(A, B) as summarized in Theorem 3.7. Finally, in section 4 we outline a technique to estimate τr(A, B) numerically to a low precision based on the property that the singular values are Lipschitz continuous over the space of matrices. The numerical results in section 5 Downloaded 04/21/14 to 212.175.32.130. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php display that at least for r close to n the singular value optimization characterization facilitate the computation of τr(A, B). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 1002 EMRE MENGI Notation. Throughout this paper A ⊗ B denotes the Kronecker product of the ×p matrices A and B, i.e., if A ∈ C ,then ⎡ ⎤ a11Ba12B ... a1pB ⎢ ⎥ ⎢ a21Ba22B ... a2pB ⎥ A ⊗ B = ⎢ . ⎥ . ⎣ . ⎦ a1Ba2B ... apB T T T T p We use the notation vec(X)=[x1 x2 ... xp ] ∈ C for the vectoral form of ×p X =[x1 x2 ... xp] ∈ C with x1,...,xp ∈ C . In our derivation we frequently benefit from the identity vec(AXB)= BT ⊗ A vec( X) for matrices A, X, B of appropriate sizes. Occasionally the notation A¯ is used to represent the conjugation of the matrix A entrywise. We reserve Ij for the identity matrix of size j × j. 2. Generalizations of the Popov–Belevitch–Hautus test. 2.1. A generalized PBH test. Here we derive an equivalent characterization C ≤ for the condition rank( (A, B)) r that facilitates the conversion of the problem (1.2) into a singular value optimization problem in the next section. The characterization is in terms of the existence of a monic matrix polynomial of A,ofdegreen − r of the form Pn−r(A)=(A − λ1I)(A − λ2I)(A − λ3I) ...(A − λn−rI) ∈ C with roots λ1,λ2,...,λn−r , satisfying a particular rank condition. The monic polynomial characterization will be derived under the assumption that the matrix A has simple eigenvalues, i.e., the algebraic multiplicities of all eigenvalues of A are one.1 In the next subsection we will remove the simple eigenvalue assumption. Suppose rank(C(A, B)) = k. Then there exists a state transformation T ∈ Cn×n such that ˜ ˜ ˜ ˜ −1 A11 A12 ˜ B1 (2.1) A = TAT = ˜ and B = TB = , 0 A22 0 k×k k×(n−k) (n−k)×(n−k) where A˜11 ∈ C , A˜12 ∈ C , A˜22 ∈ C is upper triangular, B˜1 ∈ k×m C ,andthepair(A˜11, B˜1) is controllable [5, Theorem 2.12, p. 70]. The pair (A,˜ B˜) is called the controllability canonical form for (A, B ). Since T has full rank and C(A,˜ B˜)=T C(A, B), the state transformation preserves the rank, that is, rank(C(A, B)) = rank(C(A,˜ B˜)). The uncontrollability of the system is equivalent to the existence of a left eigen- vector of A that lies in the left null space of B. This condition can be neatly expressed − ≤ − by the existence of a λ so that rank([A λI B]) n 1, which is the PBH test. More generally when rank(C(A, B)) = k, there exists a left eigenvector of A corresponding Downloaded 04/21/14 to 212.175.32.130. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1This assumption could be relaxed; it is possible to modify the derivation in this subsection for amatrixA with its controllable eigenvalues different from its uncontrollable eigenvalues. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NEAREST SYSTEMS WITH DEFICIENT REACHABLE SUBSPACES 1003 ˜ to each left eigenvector of A22 in the controllability canonical form that is contained in the left null space of B. The PBH test can be generalized based on this geometric observation. First let us suppose rank (C(A, B)) = k ≤ r, and partition A˜ and B˜ into Aˆ11 Aˆ12 Bˆ1 A˜ = and B˜ = , 0 Aˆ22 0 r×r r×(n−r) (n−r)×(n−r ) r where Aˆ11 ∈ C , Aˆ12 ∈ C , Aˆ22 ∈ C ,andBˆ1 ∈ C .NotethatAˆ22 is the lower rightmost (n − r) × (n − r) portion of A˜22 in the controllability canon- ical form (2.1).